Problem 1

Write the solution set of the given homogeneous system in parametric vector form.

(a).

Problem 2

Describe all solutions of in parametric vector form, where is row equivalent to the given matrix.

(a).

Problem 3

Mark each statement True or False.

• [ ] If is a nontrivial solution of , then every entry in is nonzero.
• [x] The equation , with and free (and neither nor a multiple of the other), describes a plane through the origin.
• [x] The equation is homogeneous if the zero vector is a solution.
• [x] The effect of adding to a vector is to move the vector in a direction parallel to .
• [ ] The solution set of is obtained by translating the solution set of .

Problem 4

Suppose has a solution. Explain why the solution is unique precisely when has only the trivial solution.

The equation has only the trivial solution .

Assume has two or more distinct solutions and

Since then meaning would have a non trivial solution. We thus arrive at a contradiction and thus must only have one unique solution.

Problem 5

Let be a matrix with two pivot positions.

(a). Does the Equation Have a Nontrivial Solution?

Yes it does have a nontrivial solution as there are 2 free variables.

(b). Does the Equation Have at Least One Solution for Every Possible ?